my hand, to the number 7, and by this process, I at length see the
number 12 arise. That 7 should be added to 5, I have certainly
cogitated in my conception of a sum = 7 + 5, but not that this sum
was equal to 12. Arithmetical propositions are therefore always
synthetical, of which we may become more clearly convinced by trying
large numbers. For it will thus become quite evident that, turn and
twist our conceptions as we may, it is impossible, without having
recourse to intuition, to arrive at the sum total or product by
means of the mere analysis of our conceptions. Just as little is any
principle of pure geometry analytical. "A straight line between two
points is the shortest," is a synthetical proposition. For my
conception of straight contains no notion of quantity, but is merely
qualitative. The conception of the shortest is therefore fore wholly
an addition, and by no analysis can it be extracted from our
conception of a straight line. Intuition must therefore here lend
its aid, by means of which, and thus only, our synthesis is possible.

Some few principles preposited by geometricians are, indeed,
really analytical, and depend on the principle of contradiction.
They serve, however, like identical propositions, as links in the
chain of method, not as principles--for example, a = a, the whole is
equal to itself, or (a+b) > a, the whole is greater than its part.
And yet even these principles themselves, though they derive their
validity from pure conceptions, are only admitted in mathematics
because they can be presented in intuition. What causes us here
commonly to believe that the predicate of such apodeictic judgements
is already contained in our conception, and that the judgement is
therefore analytical, is merely the equivocal nature of the
expression. We must join in thought a certain predicate to a given
conception, and this necessity cleaves already to the conception.